I’m about to finish my bachelors degree in Civil Engineering with an emphasis on structures at the University of Florida. I have a 3.8 Major GPA (3.7 overall I believe?), I’ve been involved in clubs, even served as president of one, have TA’d, and am involved in “research”. That’s in quotes because I’ve never known what to call it. But I work on a bridge software program that was created by my school for engineers to use. It’s used for research but I’m not really involved with the research aspect of it, moreso the development of the program but I’ve never really known what position this is. I figured it was “research assistant” because the grad students do the actual research and I’m an undergrad, so I do the grunt work necessary for them to effectively do research, I guess. But I digress. I’m currently doing a structural/geotechnical internship with a very large and well-known engineering firm.
I haven’t taken the GRE yet, but I’m studying for it now and will take it in time to make fall applications.
I haven’t compiled an entire list of schools I want to apply to, but I know that I’d like to apply to at least a few of the top schools. So UC Berkeley and Stanford are both definitely on my list.
I just want to see based off of experience and knowledge of schools at this level, what my chances of actually getting into these schools are or if there’s a more reasonable set of schools I should set my sights on. And chances of maybe getting an assistanceship or any type of aid to offset the cost of grad school.
Btw, I am applying to Master’s Programs, not going the PhD route.
If your intent is to get an MS or PhD, tuition is generally waived and you receive a stipend. Whether that is sufficient to pay for your living expenses depends on your lifestyle choices. You should contact the universities that you are targeting and ask for specifics about the level of support for grad students conducting research.
Other schools to consider:
UT Austin
University of Illinois (UIUC)
Georgia Tech
Lehigh
Your GRE score will be important, so study hard for it.
You should be able to get funding through a research assistantship.
Funny story: Before I met my husband, he got his BS in Civil at UW-Madison. His professor told him he should apply to grad school. When the man gave him a list of schools to contact, he did NOT mention UW, so my husband knew he was being straight with him. He called several schools and didn’t get much response. But when he called UT, they put him through to my father, who was head of Architectural Engineering. My dad is a friendly Texan and after talking to DH for a few minutes said, “Sure! Come on down! We’ll get you a research assistantship, no problem.” So DH moved to Austin. He was in two of my classes. Three months later, we were engaged. As of next week, we will have been married for 32 years. We have run our engineering firm out of our house for almost 19 years. 
(My dad retired from UT in 2016 after 52 years. He is 81 but still lecturing all over the world.)
If you are looking for a Master of Engineering, not research based, then many schools will expect you to self-fund. This does not mean you can’t find funding in a research group, particularly once you are in the program, but you definitely need to make sure you have a viable funding plan.
As for admission rates. The schools you list are highly selective and therefore, harder to get into particularly if you need funding. It is hard to gauge your chances but make sure you have some other choices available as suggested by other posters. I would add Purdue University as a good option for structural engineering. One of the students I have mentored was very pleased with his time there and did get funding for his MS.